Re: Incompleteness of Cantor's enumeration of the rational numbers

On 2024-11-28 12:31:35 +0000, WM said:

On 28.11.2024 12:46, Mikko wrote:

On 2024-11-27 11:12:54 +0000, WM said:
 

On 27.11.2024 10:52, Mikko wrote:

On 2024-11-26 11:05:30 +0000, WM said:
 

On 26.11.2024 10:08, Mikko wrote:

On 2024-11-25 13:55:57 +0000, WM said:

 

But before touching a rational it will touch an irrational.

 Of course as the starting point is outside of all the intervals and
every rational is in some of the intervals and therefore must be
irrational. But when it has moved to another point it has already
moved over both infinitely many irrationals

 This is true in every case. The intermediate numbers cannot be discerned. They are dark. This is so in fact between every pair of discernible real numbers: There are infinitely many dark numbers between them.

 Some of the intermediate numbers can be expressed with a finite string.

 But most cannot.
 

In particular, every rational number can.

 No. For every unit fraction there exist infinitely many smaller unit fractions, infinitely many of which cannot be expressed. They remain simply to be smaller.

 The number 0 can be expressed with a finite string. The successor of an
expresssible number can be expressed. The ratio of two expressible numbers
can be expressed. Nothing else is a rational number.

 That is potential infinity. Actual or completed infinity is a fixed quantity larger than every natural number. The set of expressed numbers however remains finite in every instance.

In mathematics everything is actual and nothing is potential. There is no
concept of "expressed", only "expressible". You can't change anything
about numbers. There are infinitely many finitely expressible numbers and
that includes all rationals.
--
Mikko